Observe that $g$, $h$ are even with respect to $x$. Then $u$ is even with respect to $x$. Since $g=0$ we conclude that $u$ is odd with respect to $t$. So basically we need to consider only $x>0,t>0$. Here teal and orange are characteristics passing through the ends of the segment $(-1,1)$ on $\{t=0\}$

1) Functions $g$, $h$ are supported in some interval $[a,b]$ (which means they are $0$ outside of it). We draw characteristics through this interval ends, we get two orange and two teal lines. They divide plane into 9 regions.

2) Observe that $u(x,t)$ is even with respect to $x$ (indeed, both $g$ and $h$ are) and odd with respect to $t$ (indeed, $g=0$; if $h=0$ we would have $u(x,t)$ even with respect to $t$). So we need to consider only $x>0$, $t>0$ and continue into 3 other quadrants by a symmetry. Instead of 9 regions we have only 4.